Propagation delay of solar wind discontinuities: Comparing different methods and evaluating the effect of wavelet denoising

Propagation delay of solar wind discontinuities: Comparing different methods and evaluating the effect of wavelet denoising

C. Munteanu,

S. Haaland,

B. Mailyan,

M. Echim,

and K. Mursula

Received 14 November 2012; revised 29 May 2013; accepted 30 June 2013; published 23 July 2013.

[

] We present a statistical study of the performance of three methods used to predict the propagation delay of solar wind structures. These methods are based on boundary normal estimations between the Advanced Composition Explorer (ACE) spacecraft orbiting the L1 libration point and the Cluster spacecraft near the Earth’s magnetopause. The boundary normal estimation methods tested are the cross product method (CP), the minimum variance analysis of the magnetic field (MVAB), and the constrained minimum variance analysis (MVAB0). The estimated delay times are compared with the observed ones to obtain a quantitative measure of each method’s accuracy. Boundary normal estimations of magnetic field structures embedded in the solar wind are known to be sensitive

to small-scale fluctuations. Our study uses wavelet denoising to reduce the effect of these fluctuations. The influence of wavelet denoising on the performance of the three methods is also analyzed. We find that the free parameters of the three methods have to be adapted to each event in order to obtain accurate propagation delays. We also find that by using denoising parameters optimized to each event, 88% of our database of 356 events are estimated to arrive within ˙2 min from the observed time delay with MVAB, 74% with CP, and 69% with the MVAB0 method. Our results show that wavelet denoising significantly improves the predictions of the propagation time delay of solar wind discontinuities.

Citation: Munteanu, C., S. Haaland, B. Mailyan, M. Echim, and K. Mursula (2013), Propagation delay of solar wind discontinuities: Comparing different methods and evaluating the effect of wavelet denoising,J. Geophys. Res. Space Physics, 118, 3985–3994, doi:10.1002/jgra.50429.

1. Introduction

[

] Due to the lack of continuous monitoring of solar wind properties close to the Earth, solar wind measurements often need to be translated from an upstream monitor to the Earth’s bow shock location. To accurately predict the propagation time of magnetic field structures embedded in the solar wind, one needs to take into account the orientation of the bound- ary normal of those structures. The challenge here is that most methods used to estimate these boundary normals are affected by small-scale fluctuations superposed on the mag- netic field structure. Instead of using frequency filtering, which smears out the discontinuities and reduces the num- ber of data points, we use wavelet denoising to reduce the effect of these fluctuations. Wavelet denoising is especially

1Institute of Space Sciences, Magurele, Romania.

2Department of Physics, University of Oulu, Oulu, Finland.

3Department of Physics, University of Bucharest, Magurele, Romania.

4Birkeland Center for Space Science, University of Bergen, Bergen, Norway.

5Max Planck Institute for Solar Systems Research, Lindau, Germany.

6Yerevan Physics Institute, Yerevan, Armenia.

7Belgian Institute for Space Aeronomy, Brussels, Belgium.

Corresponding author: C. Munteanu, Institute of Space Sciences, Atomistilor 409, P.O. Box MG-23, Bucharest-Magurele RO-077125, Romania. ([email protected])

©2013. American Geophysical Union. All Rights Reserved.

2169-9380/13/10.1002/jgra.50429

suited to remove low-amplitude high-frequency fluctuations while leaving the high-amplitude low-frequency parts of the signal unchanged.

[

] Horbury et al. [2001] used data from Advanced Composition Explorer (ACE) and Wind spacecrafts to study the propagation time of discontinuities characterized by southward interplanetary magnetic field (IMF) turnings.

They showed that the best estimates of the propagation time were obtained when the orientation of discontinuities was calculated with the cross product method (CP), which assumes that the boundary normal is given by the cross product between the mean magnetic field upstream of the discontinuity and the mean magnetic field downstream of it.

[

] Mailyan et al. [2008] studied statistically the propagation time of about 200 IMF structures between ACE and Cluster. They computed the propagation time using four different methods: the flat delay method (FD), which assumes a constant convective motion of the structure along the Sun-Earth line, the CP method, the minimum variance analysis (MVAB), and the constrained minimum variance analysis (MVAB0) finding that the best results are obtained with MVAB0.

[

] Pulkkinen and Rastätter [2009] proposed a new

method for computing the boundary normals of solar wind

discontinuities. One of their motivations was the removal

of the influence of small-scale fluctuations on the computed

normals. Their method, based on MVAB0, uses a weight

function to get smooth variations of the boundary normal,

MUNTEANU ET AL.: TIME DELAY OF SOLAR WIND DISCONTINUITIES

Figure 1. Illustration of a solar wind discontinuity propa- gating from ACE to Cluster (Target). Positions of the Sun, Earth, ACE and Cluster spacecraft, and the model disconti- nuity and its normal

at angle with respect to the solar wind velocity

are shown. Note that this is just a sketch and both position and dimensions are not to scale (adapted from Mailyan et al. [2008]).

thus effectively low-pass filtering the computed normals.

To test their time shift method, they used a global MHD model that computed the ground magnetic field. Although their method did provide some improvement in the timing of the modeled magnetic field, the improvements were not systematic and could not be detected in a statistical sample.

[

] Haaland et al. [2010] proposed a different approach of improving the boundary normal determination. They empha- sized that the filtering should be performed on the input data rather than on the obtained normals. Also, instead of frequency filtering, they suggested the use of wavelet denoising, which was already tested on magnetic field mea- surements by Haaland and Paschmann [2001].

[

] Our study extends the analysis in Mailyan et al.

[2008] by including the effect of wavelet denoising on the timing accuracy of three propagation delay estimation meth- ods: CP, MVAB, and MVAB0. We also analyze the effect of varying the free parameters of the three methods on the timing accuracy.

[

] The paper is organized as follows. Section 2 presents the data sources and database of events. Section 3 discusses the time delay estimation methods. In section 4 we describe the approach adopted for wavelet denoising. Section 5 illus- trates the results obtained for a case study, and section 6 gives the statistical results of the study. Section 7 summa- rizes the paper.

2. Data Description

[

] We use data from the ACE spacecraft in the solar wind and the Cluster spacecraft near the Earth’s magne- topause. ACE orbits the L1 libration point at approximately

km upstream of the Earth. Cluster contains four identical satellites flying in close formation around the Earth.

It has a

inclination elliptical polar orbit, with perigee at

, apogee around

, and orbital period of approxi- mately 57 h. Cluster’s apogee is in the upstream solar wind mainly from January to April every year, so our study will focus only on this period.

[

] In this work we use ACE magnetic field data from the MAG instrument [Smith et al., 1998] at 16 s resolution and solar wind velocity data from the Solar Wind Electron, Proton, and Alpha Monitor instrument [McComas et al.,

Figure 2. Sample IMF discontinuity observed by ACE and Cluster 3 on 06 January 2003: (a)

compo-

nent of the solar wind velocity measured by ACE; (b) magnetic field components measured by ACE; (c)

magnetic field components at C3 location. Figure 2b also depicts the time interval

min centered

on the discontinuity and the time intervals

min on each side of

, used in the cross product (CP)

method. In Figures 2b and 2c, blue, green, and red lines indicate

,

, and

components of the IMF.

Figure 3. Illustration of the effect of varying the parameters of time delay estimation methods on the estimation accuracy and on the quality criteria of boundary normal estimation applied on the data from the 06 January 2003 event. The first row shows the CP method results: (a) the time delay estimation accuracy

as a function of

and

, (b) the orientation angle

as a function of

and

, and (c) the shear angle

(the angle between the mean upstream and downstream magnetic field vectors) as a function of

and

. The second row shows the MVAB method results: (d) the time delay estimation accuracy

as a function of

, (e) the orientation angle

as a function of

, and (f) the eigenvalue ratio

as a function of

. The third row shows the MVAB0 results: (g) the time delay estimation accuracy

as a function of

, (h) the orientation angle

as a function of

, and (i) the eigenvalue ratio

as a function of

.

1998] at 64 s resolution. Cluster data are from the Fluxgate Magnetometer (FGM—see Balogh et al., 2001) and from the Cluster Ion Spectroscopy (CIS) experiment [Rème et al., 2001], both at 4 s time resolution. Since we focus on the propagation delay of solar wind discontinuities between two points, most of the Cluster measurements are taken only from one spacecraft, Cluster 3, from now on referred as C3.

[

] Our database consists of 356 solar wind discontinu- ities observed by both ACE and C3 in the period 2001–2012.

This database expands the data set used in Mailyan et al.

[2008]. We first identified clear magnetic field rotations in C3 measurements by visually examining the Cluster sum- mary plots. Since we focus on periods when C3 is in the upstream solar wind, we also examined the Cluster nominal position and the ion temperature (from the CIS experiment) to avoid any influences of bow shock processes (Gosling et al. [1978], Paschmann et al. [1981], and, for a recent review, Burgess et al. [2012]). Then we examined the ACE mag- netic field measurements about an hour earlier in order to see if the same field rotation could be found there. Throughout the paper we will interchangeably use the terms “magnetic field structure,” “magnetic field rotation,” “directional dis- continuity,” or just “discontinuity.” From a theoretical point of view this is not completely correct, but, for the purpose of our study, every amplitude change of at least 5 nT in less that

min in one or more components of the magnetic field is considered a discontinuity.

[

] All the satellite data were downloaded through the Automated Multi Dataset Analysis system (AMDA)

[Jacquey et al., 2010] (http://cdpp-amda.cesr.fr/DDHTML/

index.html), a web-based facility for online data analysis of space physics data.

3. Time Delay Estimation Methods

[

] IMF discontinuities are considered to be locally pla- nar structures tilted at arbitrary angles with respect to the Sun-Earth line. The configuration of a solar wind disconti- nuity propagating from ACE to C3 spacecraft is illustrated in Figure 1. The tilt of the discontinuity with respect to the flow direction, referred to as angle, and the displacement of the two satellites from the Sun-Earth line can have an important influence on the estimated time delay between the two satellites. Assuming that the radial propagation speed of the discontinuity is given by the projection of the solar wind velocity vector

onto the boundary normal direction

and that the relative distance between the two observation points with respect to the discontinuity is the observed dis- tance

projected onto

, the time delay

between the two points is given by

dt= Dn

V_SWn. (1)

We use here three boundary normal estimation methods : CP,

MVAB, and MVAB0 (as in Mailyan et al. [2008]). The CP

method assumes that the discontinuity normal is given by the

cross product between the mean upstream magnetic field

MUNTEANU ET AL.: TIME DELAY OF SOLAR WIND DISCONTINUITIES

-8 -4 0 4 8

B ACE [nT]

a) b) c)

17:15 17:31 17:46 18:02 18:17 18:33 -8

-4 0 4 8

-8 -4 0 4 8

-8 -4 0 4 8

-8 -4 0 4 8

-8 -4 0 4 8

Time [UT]

17:15 17:31 17:46 18:02 18:17 18:33 Time [UT]

17:15 17:31 17:46 18:02 18:17 18:33 Time [UT]

B ACE [nT]

d) e) f)

Figure 4. Illustration of wavelet denoising for the event on 06 January 2003. Shown are the denoising results for the Morlet wavelet function at the threshold levels (a) p

and (d) p

, the Mexican Hat wavelet at (b) p

and (e) p

, and Paul wavelet at (c) p

and (f) p

. Blue, green, and red lines indicate original (thin lines) and denoised (thick lines)

,

, and

components of the magnetic field measured by ACE.

and the mean downstream magnetic field

[Colburn and Sonett, 1966]:

n_CP= B1B2

|B₁B2|. (2)

Strictly speaking, equation (2) is valid only in case of tangential or quasi-tangential discontinuities (TDs), i.e., pla- nar structures with zero magnetic field along the normal [Colburn and Sonett, 1966]. The use of the CP method is justified by the fact that the large majority of our disconti- nuities have a small magnetic field normal component (as shown in Figure 11 along with other results). To compute the two vectors, we need to set two time intervals:

and

. The time interval

is centered on the discontinuity with its left margin coinciding with the end point of a time inter- val of length

over which the average field at “left,”

, is computed. The right margin of

coincides with the first point of a time interval of length

over which the average field at “right,-”

, is computed (see Figure 2 for a graphi- cal representation of the two time intervals). The uncertainty in boundary normal estimation increases with increasing collinearity between

and

[Knetter, 2005]. The shear angle , i.e., the angle between

and

, and the angle can be used as quality factors for the normal estimation.

Mailyan et al. [2008] showed that values of larger than

can lead to erroneous time delay estimations.

[

] MVAB is the most frequently used method to obtain the orientation of a planar magnetic field structure [see, e.g., Sonnerup and Scheible, 1998]. One first computes the eigenvectors and eigenvalues of the covariance matrix of magnetic field measurements,

:

M=hBBi–hBihBi, (3)

where

denotes averaging over a certain time interval centered on the discontinuity, indicated here as

. The eigenvector corresponding to the smallest eigenvalue is used as an estimator for the boundary normal

. The ratio

between the intermediate and minimum eigenvalues, called

, and the angle can be used as quality factors of the normal estimation [Mailyan et al., 2008].

[

] Previous studies about solar wind discontinuities sug- gest that most of them resemble TDs [Knetter, 2005]. Know- ing this, we can estimate the boundary normal using the constrained minimum variance analysis (MVAB0), where the normal magnetic field is zero by definition [Sonnerup and Scheible, 1998]. In MVAB0, the covariance matrix

(equation (3)) is replaced by

Q⁰=PikMPnjwith : Pij=ıij–bibj, (4)

where

for

and 0 otherwise, and

is the direction of the average magnetic field. The time inter- val

centered on the discontinuity and used to calculate the covariance matrix

is a free parameter in MVAB0.

The eigenvalues and eigenvectors of

have now a different meaning: the lowest eigenvalue is zero, and its correspond- ing eigenvector is

. The eigenvector corresponding to the lowest nonzero eigenvalue is now the normal estimator

. The ratio between the maximum and intermediate eigenvalues, called

, and the angle can be used as quality factors of the normal estimation [Mailyan et al., 2008].

[

] The results from the above mentioned three methods are also compared with the results obtained assuming a sim- ple convective motion of discontinuities along the Sun-Earth line, referred to in the literature as the flat delay method (FD) [Mailyan et al., 2008]. Here, the time delay between the two observation points is given by

dt_FD=Dx

Vx

, (5)

where

is the

component of the distance

between

ACE and C3 and

is the

component of the solar wind

velocity vector

.

Figure 5. Illustration of the effect of model parameters on the accuracy of time delay estimation and quality criteria for the wavelet denoised data (Morlet,

) shown in Figure 4d. The format is the same as in Figure 3.

4. Wavelet Denoising

[

] Wavelet denoising is a powerful technique bearing similarities with frequency filtering. Instead of removing frequency components from the signal, wavelet denoising removes certain wavelet coefficients based on their ampli- tude. The continuous wavelet transform of a time series

is

T(a,b) = Z 1

–1

f(t) ^a,b(t) dt, with ^a,b(t) =a^–1/2 t–b

a

, (6)

where

is the scale parameter,

is the translation param- eter, is the wavelet mother function, and

is the wavelet coefficients matrix (see, e.g., Daubechies [1992] for more details).

[

] The large-amplitude low-frequency components of the time series and the low-amplitude high-frequency ones (the “noise”) are occupying different amplitude ranges in the coefficients matrix

. Our study uses hard thresh- olding as a wavelet denoising method, in which all wavelet coefficients below a certain amplitude level are set to zero.

The threshold amplitude level

is defined here as a percent- age of the total amplitude range of the coefficients matrix.

For example, a denoising with

leaves the time series unchanged while a denoising with

sets to zero all coefficients with amplitudes smaller than 10% of the total amplitude range of the matrix

. The resulting wavelet coefficients are defined as

T^d(a,b) =

T(a,b) , if |T(a,b)| > (p/100)max(|T(a,b)|), 0 , if |T(a,b)|(p/100)max(|T(a,b)|). (7)

There is also the possibility of using soft thresholding, where all wavelet coefficients are translated toward zero by the amount

[Donoho, 1995]. The soft thresholding technique was devised in order to preserve the smoothness of the original signal after denoising. Since we

are interested only in preserving the sharp discontinuities, the hard thresholding method is more appropriate.

[

] Because the continuous wavelet transform is redun- dant, there is no unique way of defining a reconstruction for- mula. The inverse continuous wavelet transform is presented classically in the double integral form:

f^d(t) =C Z

a

Z

b

a^–2T^d(a,b) ^a,b(t) dadb, (8)

where

is a constant depending only on the wavelet mother function (empirically derived values of

for some commonly used wavelet functions are given in Torrence and Compo [1998]).

[

] An important factor to be considered in wavelet anal- ysis is the wavelet shape, which should reflect the type of features present in the time series. We tested three families of wavelets: Morlet, known to be accurate in the frequency domain, Paul that has a good time resolution, and the second derivative of a Gaussian, also known as the Mexican Hat wavelet, that has lower frequency and time resolutions [De Moortel et al., 2004]). Since our denoising thresholds are very low, i.e., only a small fraction of the signal is removed, properties such as frequency and time resolution will not have a profound effect on the reconstructed time signal. The effects of the wavelet basis on the denoising are discussed in sections 5 and 6.

[

] As used here, wavelet denoising has two free parame- ters: the wavelet function and the threshold level

. Values of

larger than

lead to the smearing out of discontinu- ities; therefore, the maximum value of

is set to 10. This is by no means the only way of denoising a time series using wavelet-based algorithms, and we plan to extend our study in subsequent papers by testing other denoising schemes.

The denoising procedure described above is applied inde-

pendently to each of the three components of the magnetic

MUNTEANU ET AL.: TIME DELAY OF SOLAR WIND DISCONTINUITIES

Figure 6. The fraction of discontinuities with time delay accuracy

within

min,

, as a function of method parameters:

for (a) CP as a function

and

, (b) for MVAB as a function

, and (c) for MVAB0 as a function of

. The parameters that maximize the 2 min fractions are

min and

min for CP, and

min and

min for MVAB and MVAB0, respectively.

field. Further details about wavelet denoising can be found in Donoho and Johnstone [1995], Donoho [1995], and Donoho et al. [1995].

5. Case Study, 6 January 2003

[

] Figure 2 shows a sample discontinuity observed by ACE at 18:01 UT on 06 January 2003. Figures 2a and 2b depict the

component of the solar wind velocity and the magnetic field measured by ACE respectively, while the magnetic field measurements of the same discontinuity detected by C3 at 19:03 UT are shown in Figure 2c. The observed time delay,

, between ACE and C3 is in this case 62 min.

[

] We have computed the time delay accuracy

, where

is the time delay estimated using one of the three methods (CP, MVAB, or MVAB0), and also the related quality factors (

,

,

, and

). The results are shown in Figure 3. The parame- ters

,

,

, and

are varied from 2 to 10 min, with a step size of 1/3 min 0.33 min. The choice of the 1/3 min (20 s)

time step is motivated mainly by the time resolution of ACE magnetic field data (16 s). If we had used an increment of 1/4 min (15 s), the time step would have been smaller than the time resolution of the data, and this would have intro- duced unnecessary and redundant computations. A step size of 20 s assures that every increment corresponds to adding at least one data point to the computations. The best result for CP is obtained when

min and

min, corre- sponding to a minimum in

of

s. We see that good accuracy (values of

smaller than 2 min) corresponds to fairly small values of

and relatively large values of the quality factor . Figure 3 shows that, in the case of this dis- continuity, values of

larger than about

and smaller than

correspond to large values of

, i.e., small accuracy of time delay estimation.

[

] Figure 3 also presents the results for

(

),

(

), and

(

) as a function of

(

). The best results are obtained for

min, which give a minimum

of

s. The sharp separation in MVAB between accurate time delay predictions for val- ues of

min and highly inaccurate ones for

min

35 40 45 50 55

f 2-min [%]

a)

20 25 30 35 40

f 2-min [%]

b)

0 1 2 3 4 5 6 7 8 9 10

35 40 45 50 55

Denoising threshold - p [%]

f 2-min [%]

c)

Figure 7.

for the three time delay estimation methods as a function of threshold amplitude level

for (a) CP, (b) MVAB, and (c) MVAB0. The method parameters used in the calculations are

,

,

,

and

. The blue, green, and red lines indicate the wavelet functions Morlet, Mexican Hat, and Paul,

respectively.

Figure 8. Distributions of time delay accuracy

for (b) CP, (c) MVAB, and (d) MVAB0 obtained using the method parameters that maximize the 2 min fractions presented in Figure 6 (Opt. dst., blue bins), the individually optimized method parameters (Opt. met., green bins), and the individually optimized denois- ing parameters (Opt. den., red bins). Also shown is the (a) histogram of time delay accuracy obtained with the FD method. The percentage of discontinuities outside the

min interval for

are shown in the top right corner for each method.

is due to the rapid decrease in the

component around 17:57 UT, which is included in estimates using

min.

MVAB0 is much less influenced by the presence of the sharp decrease in

, leading to a more smooth dependence of

and

on

. Using values larger than 6 min for the parameter

, the quality factor

increases abruptly to values larger than

, and

decreases to values smaller than 2. This shows that, in the case of this disconti- nuity, poor time delay accuracy corresponds to large values of

and small values for

.

[

] Figure 4 shows the denoised magnetic field super- imposed on the original time series for the three wavelet functions mentioned in the previous section and two thresh- old levels, p = 2 and p = 10. We see that the threshold level has a more important effect on the results than the wavelet function (see also Figure 7). We also see that denoising with p = 10, for all wavelet bases, removes the sharp decrease in

(at 17:57 UT) responsible for the erroneous time delays obtained using

min.

[

] Figure 5 shows an analysis similar to the one pre- sented in Figure 3 using the denoised time series presented in Figure 4d (Morlet wavelet function and threshold level p = 10). While denoising leaves the results of CP and MVAB0 largely unchanged, the MVAB results are signifi- cantly improved. The MVAB method applied on denoised data predicts now a correct time delay (with an accuracy within

min) for all values of the parameter

between 2 to 10 min. Denoising also increases the eigenvalue ratio

, thus allowing the use of increased lower limits for this quality factor, resulting in a better overall data quality.

6. Statistical Results

[

] Figure 6 shows how the fraction of discontinuities with accurate time delay estimation (i.e., within

min, indicated as

) varies as a function of method parame- ters, for each of the three methods. The

were computed with

,

,

, and

ranging from 2 to 10 min with the

-90 -70 -50 -30 -10 10 30 50 70 90 5

10 15 20 25 30 35 40 45

5 10 15 20 25 30 35 40 45

5 10 15 20 25 30 35 40 45

Distribution [%]

θ [deg]

-90 -70 -50 -30 -10 10 30 50 70 90 θ [deg]

-90 -70 -50 -30 -10 10 30 50 70 90 θ [deg]

-90 -70 -50 -30 -10 10 30 50 70 90 θ [deg]

-90 -70 -50 -30 -10 10 30 50 70 90 θ [deg]

-90 -70 -50 -30 -10 10 30 50 70 90 θ [deg]

a) b) c)

-40 -30 -20 -100 10 20 30 40

-40 -30 -20 -100 10 20 30 40

-40 -30 -20 -100 10 20 30 40

Δt [min]

d) e) f)

Figure 9. Illustration of the influence of wavelet denoising on the orientation angle . (top) Histograms

of angles for (a) CP, (b) MVAB, and (c) MVAB0. (bottom) Time delay accuracy

as a function of

for (d) CP, (e) MVAB, and (f) MVAB0. Blue, green, and red colors have the same meaning as in Figure 8.

MUNTEANU ET AL.: TIME DELAY OF SOLAR WIND DISCONTINUITIES

Figure 10. Illustration of the influence of wavelet denoising on the eigenvalue ratios EvR and EvR0, computed with MVAB and MVAB0, respectively. (top) Histograms of (left) EvR and (right) EvR0.

(bottom) Time delay accuracy

as a function of (left) EvR and (right) EvR0. Blue, green, and red colors have the same meaning as in Figure 8.

same time step as described in the previous section (20 s).

The parameters that maximize

are

min,

min,

min, and

min. Using this set of parameters, 45% of discontinuities are estimated within

˙2

min from the observed times for CP, 35.5% for MVAB and 47.5% for MVAB0. The accuracy of time delay esti- mation improves when a shorter

is used in CP, decreases when increasing

in MVAB, and is relatively constant when varying

for MVAB0. In each case we obtain only modest fractions of accurate time delay estimations (below 50%) regardless of the set of parameters used. A similar opti- mization procedure was used by Weimer and King [2008]

to determine the optimum set of parameters for CP and MVAB0. They found that the best results are obtained if

min,

min, and

min and that the estimation accuracies are relatively equal for the two methods. Mailyan et al. [2008] used the set of parameters:

1 = 7

min,

min, and

min and deter- mined that the best results are obtained with the MVAB0 method.

[

] Figure 7 depicts the

computed with

,

,

, and

and different wavelet functions as a function of denoising threshold level p. The wavelet functions tested are Morlet (blue), Mexican Hat (green), and Paul (red line); the threshold level

is varied from 0 to 10, with a step size of 0.1. For example, the first point of the green line in Figure 7a represents the

computed using

and

in the CP method for all discontinuities, the second point is computed by first denoising the time series used as input in CP with the Mexican Hat wavelet and a threshold

and then calculating

as above, the third point is computed with a threshold

, and so on. Figure 7 shows that the three methods are quite stable to small values of p, but a clear decreasing trend is seen as p increases. For CP and MVAB0 the decreasing trend is reduced compared with the MVAB method, and also the variability of

is slightly

lower for CP and MVAB0. A notable result is obtained for MVAB using the Morlet wavelet, where we see that up to p 3,

is clearly above the values corresponding to the other two wavelet functions; between p = 3 and p = 5 the values of

are comparable, and above p = 5 the values obtained with MVAB are lower than the ones for the other two wavelet functions. This effect may be due to the lower temporal resolution of the Morlet wavelets that are able to filter out more efficiently the high-frequency fluctuations, leading thus to the smearing out of the sharp discontinuities.

Figure 7 also shows that by using the same set of denois- ing parameters for all discontinuities, the fraction of accurate time delay estimations shows only a very small increase for small values of the threshold parameter

, if at all.

[

] In order to calculate the optimum parameters of the boundary normal estimation methods for each event indi- vidually, we varied

,

,

, and

from 2 to 10 min, with a step size of 20 s, and determined those values that mini- mize the time delay accuracy

(as in Figure 3). Figure 8 shows the distribution of

for each method in the case of optimized method parameters (Opt. met., indicated as green bins). It is clear that the percentages of accurate time delay estimations are significantly improved. As a matter of fact,

increased up to 69% for CP, 65% for MVAB, and 59%

for MVAB0.

[

] Then we determined the optimum set of denoising

parameters for each discontinuity individually. The opti-

mization procedure is similar to the one applied to optimize

the method parameters. To optimize the set of denoising

parameters, we use the optimum method parameters for each

discontinuity and then compute the time delays for different

denoising parameters. The optimum set of denoising param-

eters is the one that maximizes the prediction accuracy. The

distributions of

in this case are presented in Figure 8

as red bins indicated as “Opt. den.” and clearly show that

the

increases up to 74% for CP, 88% for MVAB, and

-5 -3 -1 1 3 5 10

20 30 40 50 60 70 80 90 100

Distribution [%]

Figure 11. Illustration of the influence of wavelet denoising on the mean magnetic field along the boundary normal,

, computed with MVAB. (top right corner) The percentage of discontinuities with

outside the

nT interval. Blue, green, and red colors have the same meaning as in Figure 8.

69% for MVAB0. The rather small improvement for CP was expected since a high accuracy was already obtained by indi- vidually optimizing the method parameters. In the case of MVAB0 the wavelet denoising increases the 2 min fraction, but a relatively large number of discontinuities still remain outside the

min interval. The main result in Figure 8 is the improvement in the case of individually optimized denoising parameters for MVAB, for which almost 99% of discontinuities are now estimated with an accuracy within

˙6

min. Figure 8 also shows the results obtained with the fixed set of parameters that maximize

(indicated as Opt. dst.) inferred from the results of Figure 6 (

,

,

, and

). We see that

for MVAB obtained using indi- vidually optimized denoising parameters is more than 50%

larger that the corresponding fraction obtained with the fixed set of parameters above.

[

] Figure 8a illustrates the results obtained with the FD method. As expected, the prediction accuracy of FD is rather poor compared with the other three methods, and only 30%

of events are predicted with a time delay accuracy

within

˙2

min.

[

] We also studied the influence of wavelet denoising on the statistics of discontinuity orientation for the three meth- ods. The results are presented in Figure 9. We have already seen in Figures 3 and 5 that the orientation angles are influ- enced by denoising for individual cases, but we now see that the distribution is not significantly affected. The time delay estimation accuracy for MVAB is improved by denoising without significantly modifying the distribution of angles.

This shows that time delay can be accurately estimated using the MVAB method if a proper preliminary denoising is per- formed, even if the plane of the discontinuity is almost paral- lel to the Sun-Earth line ( angles close to

). In the case of CP and MVAB0 methods, we see a large scatter in timing accuracy for values of larger than

, even for the opti- mum denoising case. This means that discontinuities with large angles are predicted less accurately than the ones with small angles. A similar result was reported in Mailyan et al. [2008], which concluded that an acceptable maximum value for , for an accurate normal estimation, is

.

[

] We studied also the influence of wavelet denoising on the statistics of field rotation angles calculated with the CP method. The denoising procedure has no significant influ- ence on the distribution of angles, so we decided not to show it.

[

] Figure 10 presents the influence of wavelet denois- ing on the eigenvalue ratios for MVAB and MVAB0. The EvR distribution is not significantly affected by denoising, and the time delay accuracy is improved irrespective of the EvR value, contrary to the other two cases (Opt. dst. and Opt. met.) where better results are obtained for eigenvalue ratios above

. The EvR0 distributions show that the num- ber of discontinuities with EvR0 larger than 400 is doubled after denoising, compared with the results obtained for Opt.

dst. We see that the time delay accuracy is improved after denoising irrespective of EvR0.

[

] We also studied the influence of wavelet denoising on the mean magnetic field along the boundary normal com- puted with MVAB. The results are presented in Figure 11.

Using the parameter

, we find that 59% of discontinuities are estimated to have a

component in the

nT interval and 3.9% are outside the

nT interval. Using the individ- ually optimized method parameters (Opt. met.), we find that 68% are now in the

nT interval and only 2.2% are outside the

nT interval. For the individually optimized denois- ing parameters (Opt. den.), 64% are in the

nT interval and only 1.4% are outside the

nT interval. These results show that even before denoising the majority of disconti- nuities in our database had a very small normal component of the magnetic field and denoising only slightly increases this number. The small normal component of the magnetic field indicates that most of the discontinuities in our database resemble tangential discontinuities.

7. Summary and Conclusions

[

] We have presented a statistical analysis of the per- formance of three methods (CP, MVAB, and MVAB0) to compute the propagation delay of solar wind discontinuities and the influence of wavelet denoising on this performance.

We analyzed 356 discontinuities observed by both ACE, located at L1, and C3, close to the Earth’s bow shock, between 2001 and 2012.

[

] We found that by using the fixed set of param- eters

min,

min,

min, and

min, the fraction of discontinuities estimated to arrive at C3 within

min from the observed time delay (

) is 45% for CP, 35.5% for MVAB, and 47.5% for the MVAB0 method. These results are in good agreement with the study by Mailyan et al. [2008], which also found that the best method to obtain accurate propagation delays for solar wind discontinuities is MVAB0. By tuning the method parameters for each discontinuity individually, we can deter- mine the optimum set of method parameters. We found that

increases significantly, up to 69% for CP, 65% for MVAB, and 58% for MVAB0.

[

] Wavelet denoising was used to remove small-scale

fluctuations from magnetic measurements, which are known

to influence the estimation of the orientation of a discon-

tinuity, thus affecting the time delay estimation. We found

that by using a fixed set of denoising parameters for the

entire database of discontinuities we obtain only very small

MUNTEANU ET AL.: TIME DELAY OF SOLAR WIND DISCONTINUITIES

increases of

, if any. By determining the optimum set of

denoising parameters for each discontinuity individually, we found that

increases significantly, up to 74% for CP, 88% for MVAB, and 69% for MVAB0. The fact that MVAB is the most precise method demonstrates that it is more sen- sitive to small-scale fluctuations than CP or MVAB0, and, by denoising the input signal, we can improve significantly the accuracy of time delay estimation.

[

] When the denoising is applied with a fixed set of parameters, it does not have a significant impact on the statistics of the time delays of solar wind discontinuities in our database. Nevertheless, the denoising has a clear positive effect when applied on variable time intervals as demonstrated by the results obtained for the case study presented in section 5 and by the individually optimized denoising results presented in Figure 8. The case study shows that denoising improves the accuracy of discontinuity determination and allows for an increased eigenvalue ratio threshold, resulting in better overall data quality and the inclusion of a large number of events that originally did not meet the quality criteria, thus improving the statistics.

[40] Acknowledgments. The research leading to these results has received funding from the ESA PECS Project KEEV (97049) and from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement 313038 (STORM). C.M. acknowledges the finan- cial support by the Romanian Ministry of Science and Research and by the Wallonie-Bruxelles International through a bilateral collaboration project (593) between the Institute of Space Science (Romania) and the Catholic University of Louvain, and by the Academy of Finland through the HISSI research project 128189. We acknowledge the use of the AMDA science analysis system provided by the Centre de Données de la Physique des Plasmas (CESR, Université Paul Sabatier, Toulouse).

[41] Philippa Browning thanks the reviewers for their assistance in evaluating this paper.

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